We study approximations of chance constrained problems. In particular, we consider the Sample Average Approximation (SAA) approach and discuss convergence properties of the resulting problem. A method for constructing bounds for the optimal value of the considered problem is discussed and we suggest how one should tune the underlying parameters to obtain a good approximation of the true problem. We apply these methods to a linear portfolio selection problem with returns following a multivariate lognormal distribution. In addition to the SAA, we also analyze the Scenario Approximation approach, which can be regarded as a special case of the SAA method. Our computational results indicate the scenario approximation method gives gives a conservative approximation to the original problem. Interpreting the chance constraint as a Value-at-Risk constraint, we consider another approximation replacing it by the Conditional Value-at-Risk constraint. Finally, we discuss a method to approximate a sum of lognormals that allows us to find a closed expression for the chance constrained problem and compute an efficient frontier for the lognormal case.
Submitted for publication (02/08). The first author is a PhD candidate in Mathematics at the Pontifical Catholic University of Rio de Janeiro, RJ, Brazil. The second and third authors are Professors at the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Tech, Atlanta, GA, U.S.A.